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Quantum Error-Correction Codes
Quantum Error-Correction Codes
Nothing is perfect, and everything is prone to errors. But what makes quantum information different from classical information when it comes to error correction?
Any physical system inevitably interacts with its surroundings, which are collectively referred to as the environment. These interactions have particularly severe effects on quantum systems. Quantum information is represented through a delicate state of superposition that the environment tends to knock out. This leads to decoherence and the loss of quantum information. Furthermore, quantum gates involved in quantum information processing reside in a continuum of unitary transformations, and an implementation with perfect accuracy is unrealistic for such quantum gates. Even worse, small imperfections may accumulate and result in serious errors in the state undergoing gate operations. On account of such, the errors in quantum information are clearly continuous. Detecting these continuous errors, not to mention correcting them, already seems to be a formidable task.
The principles themselves of quantum mechanics make handling quantum errors a particular challenge. In classical information processing, the basic approach involves creating duplicate copies before processing any information and comparing the output of the different copies to check for any error. For quantum information, this approach is not allowed due to the no-cloning theorem that prevents copying unknown quantum states. The measurement introduces another obstacle. In a classical case, one can probe the system and correct an error if necessary. However, this tactic does not work in quantum mechanics since the measurement disturbs the quantum states.
Amazingly, despite these apparent difficulties, it is possible to successfully correct quantum errors. This is achieved by suitably encoding quantum information. If the quantum information is encoded appropriately, then it can be recovered by merely correcting a discrete set of errors, as long as the error rate is not too high.
The Nine-Qubit Code
Bit-Flip Errors
Phase-Flip Errors
Arbitrary Errors
Quantum Error-Correction Theorems
Quantum Error-Correction Conditions
Discretization of Errors
Stabilizer Formalism
Stabilizers: Concept
Stabilizer Formalism: Overview
The Pauli and Clifford Groups
Properties of Stabilizers
Stabilizer Circuits
Examples
Stabilizer Codes
Error-Correction Conditions
Error-Syndrome Detection and Recovery
Encoding
Examples
Surface Codes
Toric Codes
Planar Codes
Recovery Procedure
Appendix
The Pauli and Clifford Groups
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